| L(s) = 1 | + (−0.125 + 0.992i)2-s + (2.00 − 1.65i)3-s + (−0.968 − 0.248i)4-s + (−0.607 − 2.15i)5-s + (1.39 + 2.19i)6-s + (−0.661 + 0.214i)7-s + (0.368 − 0.929i)8-s + (0.706 − 3.70i)9-s + (2.21 − 0.332i)10-s + (0.595 + 0.0752i)11-s + (−2.35 + 1.10i)12-s + (3.62 + 0.692i)13-s + (−0.130 − 0.683i)14-s + (−4.78 − 3.30i)15-s + (0.876 + 0.481i)16-s + (−0.812 − 3.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.0886 + 0.701i)2-s + (1.15 − 0.957i)3-s + (−0.484 − 0.124i)4-s + (−0.271 − 0.962i)5-s + (0.569 + 0.896i)6-s + (−0.250 + 0.0812i)7-s + (0.130 − 0.328i)8-s + (0.235 − 1.23i)9-s + (0.699 − 0.105i)10-s + (0.179 + 0.0226i)11-s + (−0.679 + 0.319i)12-s + (1.00 + 0.191i)13-s + (−0.0348 − 0.182i)14-s + (−1.23 − 0.853i)15-s + (0.219 + 0.120i)16-s + (−0.197 − 0.767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47401 - 0.441911i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47401 - 0.441911i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.125 - 0.992i)T \) |
| 5 | \( 1 + (0.607 + 2.15i)T \) |
| good | 3 | \( 1 + (-2.00 + 1.65i)T + (0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (0.661 - 0.214i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.595 - 0.0752i)T + (10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (-3.62 - 0.692i)T + (12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (0.812 + 3.16i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (-0.0388 + 0.0469i)T + (-3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-0.789 + 0.840i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.468 - 7.44i)T + (-28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (2.57 - 0.659i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (3.81 - 6.93i)T + (-19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (-6.21 + 5.83i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-6.59 - 9.08i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-4.24 - 10.7i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (1.34 + 0.852i)T + (22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (6.13 + 13.0i)T + (-37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (6.62 + 6.22i)T + (3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-5.59 - 0.351i)T + (66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (9.11 - 3.60i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (4.47 + 2.10i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (-5.73 - 6.92i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (4.00 + 3.31i)T + (15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (-6.90 + 14.6i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-0.0893 + 0.00561i)T + (96.2 - 12.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.52383385574062745051291876039, −11.11633043170418777434321464814, −9.357337237619928126384418185973, −8.900230746525333657372774069744, −8.053226372888577384233947460171, −7.23031220949854089151170080292, −6.14803582937553025280684438721, −4.68461731356952405999977330856, −3.24750107515057407392655767540, −1.36350753309382425959552025323,
2.39653011306907628153122358733, 3.56706958614003954800406509201, 4.09758828234590299629414448516, 6.00142780061854956531114889130, 7.53434014056670648829862568340, 8.559073085778335775415088043543, 9.337704000624030444854201232517, 10.36567619087141110562638947524, 10.81997290395029396035366436799, 11.95503185620264908752275165964
